Problem 1 - Calculus for Biology and Medicine

Question

In Problems \(1-4\), determine the sample space for each random experiment. The random experiment consisting of tossing a coin three times.

Step-by-Step Solution

Answer

The sample space is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

Step 1: Understand the Random Experiment

The experiment involves tossing a coin three times, where each toss can result in either a 'Head' (H) or a 'Tail' (T). Our goal is to list all the possible outcomes of this experiment.

Step 2: Identify the Number of Trials

Since the coin is tossed three times, we will have three trials in this random experiment. Each trial results in either an H or a T.

Step 3: Construct the Sample Space

To construct the sample space, list all possible sequences of outcomes for the three tosses. For each of the three positions, there are two options (H or T), leading to a total of \(2^3 = 8\) possible outcomes.

Step 4: List All Possible Outcomes

Each trial is independent, so we can list the combinations as follows:- First toss: H, second toss: H, third toss: H, resulting in HHH- HHT, HTH, HTT, THH, THT, TTH, TTT Thus, the sample space is \[ \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \].

Key Concepts

Understanding Random Experiments

A random experiment is a process or activity that produces one of several possible outcomes, where the outcome cannot be predicted with certainty ahead of time. It forms the basis of probability theory. In the context of the provided exercise, the random experiment is tossing a coin three times. During each toss, you do not know which face will land face up—a 'Head' (H) or a 'Tail' (T). This unpredictability is what makes it random. In any random experiment, especially with multiple trials like in this case, identifying its structure is essential for determining outcomes and probabilities.

The Basics of a Coin Toss

A coin toss is one of the simplest forms of random experiments. Tossing a fair coin means each possible outcome, 'Head' (H) or 'Tail' (T), has an equal likelihood of happening, generally assumed to be 50%. In a single toss, two outcomes are possible. However, when a coin is tossed multiple times, these outcomes multiply, creating more complex scenarios. In our three-toss scenario, each toss can provide an H or a T, leading to numerous combinations of results. The beauty of the coin toss lies in its straightforward yet illustrative nature of fundamental probability concepts.

Exploring Probability

Probability offers a way to quantify the likelihood of an outcome of a random experiment. For a single coin toss, the probability of getting a 'Head' or a 'Tail' is both \(\frac{1}{2}\). When determining probability across multiple tosses:

• The probability of all outcomes in the sample space occurring equals 1.
• For our exercise, each specific outcome, such as HHH, has \(\frac{1}{8}\) probability because there are 8 equally likely outcomes.

Understanding probability helps in predicting outcomes, assessing risks, and making informed decisions in uncertain scenarios.

Concept of Independent Trials

In probability, independent trials refer to scenarios where the outcome of one trial does not affect the outcome of another. This is crucial, especially in coin tosses. Each toss is an independent event, meaning the result of the first toss does not influence the second. In our experiment involving three coin tosses, each toss remains unaffected by the others. This independence allows us to multiply the probabilities of each individual event when calculating the probability of a series of outcomes. Therefore, understanding independence is key to analyzing this and other similar experiments accurately.